Schauder estimates for a homogeneous Dirichlet problem in a half-space of a Hilbert space (Q5935783)

Let \(H\) be a real separable Hilbert space with inner product \((.,.)\). The author is concerned with the following infinite-dimensional Dirichlet problem: NEWLINE\[NEWLINE\lambda \psi(x)=\tfrac{1}{2}\text{Trace}[QD^2\psi(x)]=f(x)\quad (x\in H_+, \lambda >0),NEWLINE\]NEWLINE NEWLINE\[NEWLINE \psi(z)=0 \qquad (z\in \partial H_+),NEWLINE\]NEWLINE where \(H_+=\{x\in H : (x, e)>0\) with \(e\in H\}\). The function \(f\) is real and defined on \(H_+.\) In the paper, the existence and uniqueness of solutions are studied. Moreover, an optimal regularity result of Schauder-type is given.